On Dynamic Algorithms for Algebraic Problems

In this paper, we examine the problem of incrementally evaluating algebraic functions. In particular, if f(x1, x2, …, xn) = (y1, y2, …, ym) is an algebraic problem, we consider answering on-line requests of the form "change input xi to value v" or "what is the value of output yj?" We first present lower bounds for some simply stated algebraic problems such as multipoint polynomial evaluation, polynomial reciprocal, and extended polynomial GCD, proving an &#x03A9(n). lower bound for the incremental evaluation of these functions. In addition, we prove two time-space trade-off theorems that apply to incremental algorithms for almost all algebraic functions. We then derive several general-purpose algorithm design techniques and apply them to several fundamental algebraic problems. For example, we give an O( √ n  ) time per request algorithm for incremental DFT. We also present a design technique for serving incremental requests using a parallel machine, giving a choice of either optimal work with respect to the sequential incremental algorithm or superfast algorithms with O(log log n) time per request with a sublinear number of processors

Sheet-metal press line parameter tuning using a combined DIRECT and Nelder-Mead algorithm

It is a great challenge to obtain an efficient algorithm for global optimisation of nonlinear, nonconvex and high dimensional objective functions. This paper shows how the combination of DIRECT and Nelder-Mead algorithms can improve the efficiency in the parameter tuning of a sheet-metal press line. A combined optimisation algorithm is proposed that determines and utilises all local optimal points from DIRECT algorithm as Nelder-Mead starting points. To reduce the total optimisation time, all Nelder-Mead optimisations can be executed in parallel. Additionally, a Collision Inspection Method is implemented in the simulation model to reduce the evaluation time. Altogether, this results in an industrially useful parameter tuning method. Improvements of an increased production rate of 7% and 40% smoother robot motions have been achieved

Optimization Of Groundwater Remediation With New Efficient Parallel Algorithm For Global Optimization

Application of optimization algorithm to PDE modeling groundwater remediation can greatly reduce remediation cost. However, groundwater remediation analysis requires a computational expensive simulation, therefore, effective parallel optimization could potentially greatly reduce computational expense. The optimization algorithm used in this research is Parallel Stochastic radial basis function. This is designed for global optimization of computationally expensive functions with multiple local optima and it does not require derivatives. In each iteration of the algorithm, an RBF is updated based on all the evaluated points in order to approximate expensive function. Then the new RBF surface is used to generate the next set of points, which will be distributed to multiple processors for evaluation. The criteria of selection of next function evaluation points are estimated function value and distance from all the points known. Algorithms created for serial computing are not necessarily efficient in parallel so Parallel Stochastic RBF is different algorithm from its serial ancestor. The application for two Groundwater Superfund Remediation sites, Umatilla Chemical Depot, and Former Blaine Naval Ammunition Depot. In the study, the formulation adopted treats pumping rates as decision variables in order to remove plume of contaminated groundwater. Groundwater flow and contamination transport is simulated with MODFLOW-MT3DMS. For both problems, computation takes a large amount of CPU time, especially for Blaine problem, which requires nearly fifty minutes for a simulation for a single set of decision variables. Thus, efficient algorithm and powerful computing resource are essential in both cases. The results are discussed in terms of parallel computing metrics i.e. speedup and efficiency. We find that with use of up to 24 parallel processors, the results of the parallel Stochastic RBF algorithm are excellent with speed up efficiencies close to or exceeding 100%

A trivariate interpolation algorithm using a cube-partition searching procedure

In this paper we propose a fast algorithm for trivariate interpolation, which is based on the partition of unity method for constructing a global interpolant by blending local radial basis function interpolants and using locally supported weight functions. The partition of unity algorithm is efficiently implemented and optimized by connecting the method with an effective cube-partition searching procedure. More precisely, we construct a cube structure, which partitions the domain and strictly depends on the size of its subdomains, so that the new searching procedure and, accordingly, the resulting algorithm enable us to efficiently deal with a large number of nodes. Complexity analysis and numerical experiments show high efficiency and accuracy of the proposed interpolation algorithm